A well-attended meeting, called "Brainstorming discussion for a possible biomedical facility at CERN", was held by the European Organization for Nuclear Research (CERN) at the European Laboratory for Particle Physics on 25 June 2012. This was concerned with adapting an existing, but little used, 78-m circumference CERN synchrotron to deliver a wide range of ion species, preferably from protons to at least neon ions, with beam specifications that match existing clinical facilities. The potential extensive research portfolio discussed included beam ballistics in humanoid phantoms, advanced dosimetry, remote imaging techniques and technical developments in beam delivery, including gantry design. In addition, a modern laboratory for biomedical characterisation of these beams would allow important radiobiological studies, such as relative biological effectiveness, in a dedicated facility with standardisation of experimental conditions and biological end points. A control photon and electron beam would be required nearby for relative biological effectiveness comparisons. Research beam time availability would far exceed that at other facilities throughout the world. This would allow more rapid progress in several biomedical areas, such as in charged hadron therapy of cancer, radioisotope production and radioprotection. The ethos of CERN, in terms of open access, peer-reviewed projects and governance has been so successful for High Energy Physics that application of the same to biomedicine would attract high-quality research, with possible contributions from Europe and beyond, along with potential new funding streams.